Set reconciliation, where two parties each holding a large set of elements aim to identify their set difference, is a fundamental task in many areas. Time (computation cost) and space (communication cost) are two important metrics in this problem, and much existing work optimizes one at the expense of the other, creating a clear trade-off. We study this trade-off through algorithmic upper bounds and matching space lower bounds. On one side, we propose XYZ-Sketch. For sufficiently large $d$, XYZ-Sketch achieves high-probability reconciliation for sets whose differing elements are uniformly random, using only $(1+\varepsilon)d$ elements for communication, while achieving $O(\log(1/\varepsilon))$ insertion time and expected $O(d\log V\log^{3/2}(1/\varepsilon))$ decoding time, where $d$ and $V$ denote the size of the difference between two sets and the universe size, respectively. On the other side, we establish a broad fixed-support canonical model and prove a matching space lower bound, establishing the asymptotic optimality of XYZ-Sketch. Our proof also establishes an extremality theorem resolving Walzer's conjecture. Experiments demonstrate the practical advantages of XYZ-Sketch, and the source code is available at https://github.com/djwj233/XYZ-Sketch.

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